quantum accuracy-size trade-off.
White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/99491/
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Article:
Busch, Paul orcid.org/0000-0002-2559-9721, Loveridge, Leon Deryck and Miyadera, Takayuki (2016) Approximating relational observables by absolute quantities : a quantum accuracy-size trade-off. Journal of Physics A: Mathematical and Theoretical. 185301. ISSN 1751-8113
https://doi.org/10.1088/1751-8113/49/18/185301
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arXiv:1510.02063v3 [quant-ph] 24 Mar 2016
quantities: a quantum accuracy-size trade-off
Takayuki Miyadera
Department of Nuclear Engineering, Kyoto University, Kyoto daigaku-katsura, Nishikyo-ku, Kyoto, Japan 615-8540
E-mail: [email protected]
Leon Loveridge
Department of Computer Science, University of Oxford, Wolfson Building, Parks Rd, Oxford, UK OX1 3QD
E-mail: [email protected]
Paul Busch
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD
E-mail: [email protected]
Abstract. The notion that any physical quantity is defined and measured relative to
a reference frame is traditionally not explicitly reflected in the theoretical description of physical experiments where, instead, the relevant observables are typically represented as “absolute” quantities. However, the emergence of the resource theory of quantum reference frames as a new branch of quantum information science in recent years has highlighted the need to identify the physical conditions under which a quantum system can serve as a good reference. Here we investigate the conditions under which, in quantum theory, an account in terms of absolute quantities can provide a good approximation of relative quantities. We find that this requires the reference system to be large in a suitable sense.
Published in: J. Phys. A: Math. Theor. 49185301 (2016)
1. Introduction
Symmetry plays a fundamental role in constructing and understanding physical theories [1, 2]. It also constrains the relationship between theoretical terms in a given formalism and the world those terms are used to describe. In a theory with (say, a gauge) symmetry, the quantities which may be measured (the observables) must be invariant with respect to the relevant symmetry transformations [3, 4]. Yet, it appears that in certain circumstances, measurements of symmetric systems can be well described in terms of non-invariant quantities. Typically, the invariant quantities are relative observables of a system plus reference, whereas the non-invariant quantities provide a simplified, “absolute” description of the situation in terms of the variables of the system alone.
Problems of this kind appear naturally in investigations of the universality of quantum mechanics and the interface between quantum and classical systems. For instance, in [5–8] it is argued that the notion of time appears from stationary observables in a composite system of the system and the clock. Another example, closer to the type of situation we will consider, arises in the theory of superconductivity: though the “absolute” phase in a superconductor does not represent an observable quantity, the Josephson effect shows that the relative phase between two systems can be measured, since the relative phase operator is gauge (phase-shift) invariant. If one of these systems is large in a suitable sense, the fluctuation of the phase can be neglected and the phase can be treated effectively as a classical quantity. In this case the (statistics of a) relative phase measurement can be well approximated by the statistics of a non-invariant phase observable of a subsystem. The large system is then viewed as a (classical) reference system.
The possibility of such an approximation is considered in various forms in the literature and is accompanied by various interpretations. In [3], a review on quantum reference frames and their use as resources for information processing, it is shown that any self-adjoint operator A can be “relativised” to give an (invariant) observable U(A)
acting in the Hilbert space of a larger system composed of the original system and a reference system (henceforth we use the term system-plus-reference). Then A may approximate U(A) well if the reference system has a localisable phase-like quantity [4].
Ais viewed as an “approximator” for the invariant observableU(A) in the high reference
phase localisation limit. The large reference requirement also appears in the Wigner-Araki-Yanase theorem [9–15] as a large spread in the apparatus part of a conserved quantity, and as an asymmetric resource state in [16], where the measurement of a non-invariant operator is viewed as being simulated by the measurement of the non-invariant observable of system-plus-reference. The symmetry constraint and reference-system size also arise in work concerning superselection rules [3, 4, 17, 18].
The purpose, therefore, of this paper, is to prove that the operational “distance” between an arbitrary effect (operator) of some quantum system and the restriction of a symmetry-invariant effect of system-plus-reference depends explicitly on the spread of the symmetry generator in the state of the reference system. For good approximation, the reference frame system must be large. Our approach is quantitative and has a clear operational meaning, applies to unsharp as well as sharp observables, and holds for general continuous symmetry groups. We find, moreover, that the trade-off relations established are rooted in the uncertainty relations for certain incompatible quantities, and that therefore the size-versus-accuracy trade-offs are genuinely quantum constraints. After reviewing standard background material in Section 2, we briefly introduce in Section 3 the notion of “restriction” of operators on the system-plus-reference to operators on the system, which arises by fixing a state of the reference. Section 4 contains the relativisation map introduced in [4], accompanied by a discussion of the role of reference system localisation. Section 5 brings our main result: a trade-off between precision and size. This is given as a quantitative relation for the distance between arbitrary and restricted invariant effects; we show that the distance may be made small only when the reference state is delocalised with respect to the symmetry generator or, where applicable, localised with respect to its conjugate quantity. The origin of the trade-off as arising from quantum incompatibility is discussed, and we conclude in Section 6 with a summary and remarks on future work.
2. Mathematical preliminaries
We briefly present some mathematical concepts and notation used throughout the paper. Any Hilbert space H we consider will be finite-dimensional and we write dimH for the dimension of H. The standard inner product is denoted h · | · i:H × H → C (taken to be linear in the second argument). The algebra of bounded linear operatorsA:H → H will be denoted by L(H), with the positive operators (A≥0) given as those A∈ L(H) for which hϕ|A|ϕi ≥0 for allϕ ∈ H(ifA−B ≥0 we write A≥B orB ≤A). States in H, (equivalently, on L(H)) are given by the positive operators ρin L(H) for which the trace tr [ρ] = 1 (equivalently, positive linear functionalsω : L(H)→C with ω(1) = 1),
the convex set of which is written S(H). The extreme elements of S(H)—the pure states—satisfy ρ2 =ρ, and are given by rank one projections, or simply as unit vectors in H.
An observable E is a normalised, positive operator valued measure (pom) E :F →
L(H) on some measurable space (Ω,F) (so that E(X) ≥ 0, E(Ω) = 1, and E satisfies
the additivity property of measures). Any operator E(X) in the range of E is an effect, that is, a positive operator A for which 0≤A≤ 1; we write E(H) for the (convex) set
quantified by Dω(A, B) := |ω(A)−ω(B)|. A (state-independent) distance function D is then obtained by taking the supremum over all states:
D(A, B) := sup ω
Dω(A, B) = sup ω |
ω(A)−ω(B)| =kA−Bk.
Since ω(A) andω(B) are probabilities, D has a clear operational meaning [20].
Let G be a group. A system is said to possess G as a symmetry (group) if there is a projective (anti-)unitary representation {U(g)}g∈G of G on H. That is, the U(g) are (anti-)unitary operators satisfyingU(g)U(g′) =µ(g, g′)U(gg′) with coefficients
|µ(g, g′)| = 1 The projective representation induces a ∗-automorphism αg : L(H) →
L(H) byαg(A) =U(g)AU(g)∗, which satisfiesαg◦αg′ =αgg′. In this paper, we consider a finite-dimensional connected Lie group G, in which case each U(g) is automatically unitary. We denote by g the Lie algebra of G. There exists a neighbourhood O of e (the unit of G) such that every element g ∈O is uniquely expressed as g =exp(n) with n ∈ g. Once we fix n ∈ g, we can define a line g(s) = exp(sn) with a variable real parameter s. This satisfies g(s)g(t) = g(s+t) for small enough |s| and |t|. U(g) can be set so as to satisfy U(g(s))U(g(t)) =U(g(s+t)). Then Stone’s theorem on unitary representations of one-parameter Abelian groups applies, to obtain U(esn) =eisN with some self-adjoint operator N (a generator) on the Hilbert space. We say an effect A is G-invariant if U(g)AU(g)∗ =A for all g ∈ G. This naturally implies [N, A] = 0 for each generator N. Similarly, a stateρ is (G-)invariant ifU(g)∗ρU(g) =ρ for all g ∈G, equivalently if [N, ρ] = 0, or if ω =ωρ : L(H) → C satisfies ωρ(A) = ωρ(U(g)AU(g)∗) for all A∈ L(H) and g ∈G.
Given Hilbert spaces H and K (as “output” and “input space”, respectively), we consider channels in the Heisenberg picture. A linear map Λ :L(H)→ L(K) is called a
channelif (i) Λ(1) =1and (ii) Λ⊗id:L(H)⊗C
d→ L(K)⊗Cdis positive for alld(that
is, Λ is completely positive). In the Schr¨odinger picture a state ρ∈ S(K) is mapped to Λ∗(ρ)∈ S(H), which is defined by tr
ρΛ(A)
= tr
Λ∗(ρ)A
for allA∈ L(H). If a group Ghas (projective) representationsU1, U2 onHandK, respectively, a channel Λ is called (G-)covariant if it satisfies Λ U1(g)AU1(g)∗
=U2(g)Λ(A)U2(g)∗ for all A ∈ L(H) and
g ∈G. G-covariant channels map invariant effects to invariant effects.
3. Restriction map
Let us consider a system S and a reference R with Hilbert spaces HS and HR, respectively. The Hilbert space of the total system (system-plus-reference) is H := HS ⊗ HR. As a preliminary observation we note that given a self-adjoint A∈ L(H)≡
through which one can “view” effects ofHS. More generally, fixing a stateωR ∈ S(HR) we define the restriction map ΓωR :L(H)→ L(HS) via
ΓωR(A⊗B) =AωR(B),
extended by linearity to the whole of L(H). For any ωR, ΓωR is unital and completely positive. Of course, ΓωR(A⊗1) = A. Conversely, any channel Γ : L(H) → L(H
S) satisfying Γ(A⊗1) =Ahas a unique state ω
R onHR satisfying Γ(A⊗B) =AωR(B).‡ Physically, ΓωR may be understood as providing a reduced description in terms of S alone, contingent upon the state of the reference being ωR. The Schr¨odinger picture is helpful to see the physical meaning of the restriction map. It maps a state ω on the system to ω ◦Γ = ω ⊗ωR. Thus Γ describes the usual “tracing out” operation for product states. Note that the restriction map is not G-covariant in general for non-invariant states ωR.
We now state our problem. Let G (a connected Lie group) be a symmetry of the system and the reference; that is, there exist projective unitary representations US and URonHS andHRrespectively, with the tensor product representation writtenUS⊗UR (i.e., (US⊗UR)(g) =US(g)⊗UR(g) for eachg ∈G). For each (possibly non-invariant) A ∈ E(HS), how close can Γ(E) be to A by choosing a globally invariant E ∈ E(H) (that is, an E for which (US(g)⊗UR(g))E(US(g)⊗UR(g))∗ = E for all g ∈ G) and restriction map Γ?
We begin by considering the special case of invariant E obtained by relativisation, before addressing the general case.
4. Relativisation map
In [4] it is shown that poms, and therefore self-adjoint operators and effects, can be
“relativised” to give associated invariant (with respect to some chosen groups) poms,
self-adjoint operators and effects in a new Hilbert space. For instance, position relativises to relative position, angle to relative angle, and phase (as apom) to relative phase. We
therefore view the relativisation procedure as giving rise to relative observables and effects even when defined on arbitrary poms (or effects). Here we study relativisation
for the case (of representations of) G = U(1) (otherwise called the phase group) to give general quantitative tradeoff relations between (in)accuracy and reference system localisation.
Specifically, let NS and NR denote number operators acting inHS and HR. Thus, for example, NS is a self-adjoint operator onHS with orthonormal basis of eigenvectors {|ni}n∈I ⊂ HS, I = {0, ..., N0 −1}, so that NS = Pnn|nihn| ≡
P
nnPn. We define
‡ To see this, we introduce a sesquilinear map L(H)× L(H) → L(HS) by hX, Yi := Γ(X∗Y)− Γ(X∗
)Γ(Y). It satisfies a Cauchy-Schwarz-type inequalilty: hX, YihY, Xi ≤ khY, YikhX, Xias shown in Section 5 (Lemma 3). Since hA ⊗1, A⊗1i = 0, we have hA⊗1,1⊗Bi = 0. It implies
Γ(A⊗B) =AΓ(1⊗B) and Γ(A⊗B) = Γ(1⊗B)Afor all AandB. Thus Γ(1⊗B) is proportional
N := NS + NR on H = HS ⊗ HR, and denote the associated unitary groups by US(θ) =eiNSθ, U
R(θ) =eiNRθ and U(θ) =eiN θ =US(θ)⊗UR(θ);θ ∈[−π, π).
Stipulating that only invariant, or relative effects are observable [4], we may then only measure those effects E ∈ E(H) for which U(θ)EU(θ)∗ = E or, equivalently, [E, N] = 0. We may construct relative effects in E(H) =E(HS ⊗ HR) out of arbitrary effects inE(HS); first we introduce the linear mapping (an adapted and generalised form of that given in [3]) U:L(HS)→ L(H) (see below for the full definition):
U(A) =
Z π
−π
US(θ)AUS(θ)∗⊗F(dθ).
Here F is a covariant phase pom on the reference (i.e., acting in HR), defined as any pom on (the Borel algebraB [−π, π) of) [−π, π) for which
UR(θ)F(X)UR(θ)∗ =F(X∔θ),
where ∔ denotes addition modulo 2π.
Here we give the definition of U(A) and investigate its properties. For a covariant
phase pomF there exists [22] a uniquely determined positive operator T ≥0 satisfying
for all measurable subsets X,
F(X) = 1 2π
Z
X
dθUR(θ)T UR(θ)∗,
with 1 2π
Rπ
−πdθ UR(θ)T UR(θ)∗ = 1. That is, for each state ωR, there exists a positive
smooth function fωR(θ) :=
1
2πωR(UR(θ)T UR(θ)∗) satisfying ωR(F(X)) =
R
Xdθ fωR(θ). We denote the absolutely continuous measureωR◦FbyµF
ωR, so thatµ F
ωR(dθ) = fωR(θ)dθ.
U(A) is defined by
U(A) = 1
2π
Z π
−π
dθUS(θ)AUS(θ)∗⊗UR(θ)T UR(θ)∗.
It is easy to see that U(A) is G-invariant and becomes an effect for any effect A.
Again, the Schr¨odinger picture is helpful to see the physical meaning of U(A). In
the Schr¨odinger picture, the state must be invariant while arbitrary effects are allowed. In this picture the simplest (thus product) relative effect has the formA⊗T. Switching to the Heisenberg picture, we obtain U(A).
ΓωR(U(A)) is given by
ΓωR(U(A)) =
Z
µF
ωR(dθ)US(θ)AUS(θ) ∗.
Quantifying the distance between an effect A ∈ E(HS) and a restriction of a relativised version, viz. ΓωR(U(A)), amounts to estimating the following quantity:
D A,ΓωR U(A)
=
Z π
−π µF
ωR(dθ) e
iθNSAe−iθNS
−A
.
For 0 ≤ ǫ < 1, the overall width at confidence level 1−ǫ of the measure µF
ωR is defined by
Wǫ(µF
ωR) := inf
w∃θ : µFωR I(θ, w)
where I(θ, w) denotes the closed interval of width w ≤ 2π centred at θ in [−π, π), understood as a circle. It can be shown (see, e.g., [23, Chapter 12]) that the above infimum is actually a minimum. Hence, due to the absolute continuity of F, there is a θ0 such that µF
ωR I(θ0, Wǫ(µ F
ωR)
= 1−ǫ. We call I(θ0, Wǫ(µF
ωR)) an ǫ-support of the measure µF
ωR (noting that it need not be unique). In addition, we define another quantity W0
ǫ(µFωR)—the overall width around 0—by
Wǫ0(µF
ωR) := inf
w µFω
R I(0, w)
≥1−ǫ . Of course, W0
ǫ(µFωR)≥Wǫ(µ F
ωR) holds. Note that this inf can also be replaced by min. That is, we have
µF
ωR I(0, W
0
ǫ(µFωR))
= 1−ǫ. (2)
Proposition 1. Let U be a relativisation map and ΓωR a restriction map. For an
arbitrary effect A and 0≤ǫ <1, it holds that
D A,ΓωR(U(A))
≤
[NS, A]
1
2W 0
ǫ(µFωR)(1−ǫ) +πǫ
.
Proof. Letθ ∈[−π, π). d
dθe iNSθ
Ae−iNSθ
=eiNSθ
i[NS, A]e−iNSθ
is used to obtain
eiNSθAe−iNSθ
−A=
Z θ
0
dθ′eiNSθ′i[N
S, A]e−iNSθ ′
.
Taking the norm yields
eiNSθAe−iNSθ−A ≤ |θ|
[NS, A] .
Thus,
D A,ΓωR(U(A))
≤
[NS, A]
Z π
−π µF
ωR(dθ)|θ|
=[NS, A]
Z Wǫ0(µ
F
ωR)/2
−W0
ǫ(µFωR)/2
µF
ωR(dθ)|θ|
+
Z π
W0
ǫ(µFωR)/2
µFωR(dθ)|θ|+
Z −Wǫ0(µFωR)/2
−π
µFωR(dθ)|θ|
!
≤[NS, A]
12Wǫ0(µFωR)(1−ǫ) +πǫ
,
where we used Eq. (2).
Therefore, we may conclude that for ωR with strong localisation aroundθ = 0, the discrepancy between A and ΓωR(U(A)) is small.
AnωRwhich has small overall width but large overall width around 0 does not give good approximation of A by U(A). However, we can construct a deformed quantity Uθ0(A) forθ0 ∈[−π, π) as
Uθ
0(A) = (1⊗U
R(θ0))U(A)(1⊗U
which is G-invariant. Recall that for 0 ≤ ǫ < 1, each ωR has θ0 such that µF
ωR(I(θ0, Wǫ(µ F
ωR))) = 1−ǫ. For such a θ0, we have
D A,ΓωR(Uθ0(A))
≤
[NS, A]
1
2Wǫ(µ
F
ωR)(1−ǫ) +πǫ
.
In the following we show that the high localisation ofωR is necessary to haveAwell approximated by ΓωR(U(A)). In this illustration we assume that NS has eigenstates|0i and |1i with respective eigenvalues 0 and 1. We consider a particular effect A defined by A = 12 |1ih1|+|0ih0|+|0ih1|+|1ih0|
. The effect A does not commute with NS. We show that the magnitude of difference between A and ΓωR(U(A)) is related to the localisation property indeed.
Proposition 2. For A= 1
2 |1ih1|+|0ih0|+|0ih1|+|1ih0|
, it holds that
D A,ΓωR(U(A))
≥ 2ǫ1−cos 12Wǫ0(µF
ωR)
Proof. As eiNSθAe−iNSθ−A= 1
2
|0ih1|(e−iθ−1) + (eiθ −1)|1ih0|
holds, we have
D A,ΓωR(U(A))
= 1 2
|0ih1|c+c|1ih0|
= 12|c|,
where c:=R
µF
ωR(dθ)(1−e
iθ). This|c| is estimated fort∈R as
|c| ≥Re c=
Z
µF
ωR(dθ)(1−cosθ)
=
Z
1−cosθ>t µF
ωR(dθ)(1−cosθ) +
Z
1−cosθ≤t µF
ωR(dθ)(1−cosθ)
≥t·µF
ωR {θ|1−cosθ > t}
=t
1−µF
ωR {θ|1−cosθ ≤t}
.
We put t = 1−cos 1 2W
0 ǫ(µFωR)
to obtain (using again (2))
t
1−µF
ωR {θ|1−cosθ≤t}
=
1−cos 1 2W
0 ǫ(µFωR)
ǫ.
Thus we can see that as Wǫ0(µFωR) becomes large so does the discrepancy
D A,ΓωR(U(A))
.
General effects can be treated in essentially the same manner. Suppose A does not commute with NS = P
nnPn, where each Pn is a projection. A can be decomposed into two parts, as A = P
nPnAPn +
P
n6=mPnAPm, and therefore eiNSθAe−iNSθ−A=P
n>m((ei(n−m)−1)PnAPm+ (e−i(n−m)−1)PmAPn) holds. Putting cnm:=R
µF
ωR(dθ)(1−e
i(n−m)θ), we have
D A,ΓωR(U(A))
= X n>m
cnmPnAPm+cnmPmAPn
.
Pm|mi=|miso that|φi:= √1
2(|ni+|mi) satisfieshφ| (cnmPnAPm+cnmPmAPn)|φi= 1
2(cnmhn|A|mi+cnmhm|A|ni) =|cnm|| hn|A|mi | 6= 0. Thus we have a bound, D A,ΓωR(U(A))
≥ |cnm|| hn|A|mi |. |cnm| is bounded as
|cnm| ≥Recnm ≥
Z n−πm
− π n−m
µFωR(dθ)(1−cos(n−m)θ)
≥t·µFωR
{θ|1−cos(n−m)θ > t} ∩[− π n−m,
π n−m]
.
For sufficiently large ǫ satisfying W0
ǫ(µFωR) ≤
2π
n−m, we put t = 1−cos n−m
2 W 0 ǫ(µFωR)
to obtain
t·µFωR
{θ|1−cos(n−m)θ > t} ∩[− π n−m,
π n−m]
=
1−cos
n−m
2 W
0 ǫ(µFωR)
ǫ.
Thus D A,ΓωR(U(A))
can be bounded by a similar inequality in which W0
ǫ(µFωR) plays a role. In the following we treat only the special effect A = 1
2(|0ih0|+|1ih1|+|0ih1|+ |1ih0|) for simplicity.
In [4] it is shown that a large reference frame allows for good phase localisation around 0. Now we show that to attain good localisation in the sense of small overall width, a large Hilbert space dimension for the reference is needed.
Lemma 1. The overall width W0
ǫ(µFωR) of µ F
ωR around 0 satisfies, for any ǫ≥0, 1−ǫ≤ dimHR
2π W 0 ǫ(µFωR).
Proof. For anyX ⊂[−π, π) and for any stateωR, we haveωR(F(X))≤tr[F(X)], where tr denotes the trace in HR. Now for any eigenstates |ni ofNR, we have hn|F(X)|ni= |X|/2π and so tr[F(X)] = dimHR|X|/2π. We put X = [−W0
ǫ(µFωR)/2, W
0
ǫ(µFωR)/2] to obtain the result (using once more the equality (2)).
Note that this estimate is not tight. For instance, for ǫ= 0 we obtain W0
0(µFωR)≥
2π
dimHR. However, one can show that for any stateW
0
0(µFωR) = 2π. That is, no state can be strictly localised [24]. In fact, if we were to assume that a stateωR could be strictly localised, there would exist an open setX ⊂[−π, π) for which ωR(F(X)) = 0. Defining f(θ) := ωR eiNRθ
F(X)e−iNRθ and noting that the Hilbert space is finite dimensional, this function can be analytically continued to the whole complex plane. This analytic function, by assumption, satisfiesf(θ) = 0 on some open interval ofR. This means that f(θ) = 0 on the whole plane, which contradicts ωR F([−π, π))
= 1.
Proposition 3. Let A = 1
2 |1ih1|+|0ih0|+|0ih1|+|1ih0|
. For any ǫ≥0,
D A,ΓωR(U(A))
≥ 12
1−cos
π 1−ǫ dimHR
Proof. We use Lemma 1 and Proposition 2 to obtain cos W0
ǫ(µFωR)/2
≤ cos (π(1−ǫ)/dimHR).
In particular for ǫ= 12, we obtain
D A,ΓωR(U(A))
≥ 1 4
1−cos
π 2 dimHR
,
which is a non-trivial bound for dimHR <∞.
Let us estimate the overall width around 0 in terms of the standard deviation of the number operator ∆ωRNR := (ωR(N
2
R)−ωR(NR)2)1/2. We obtain the following bound for W0
ǫ(µFωR).
Lemma 2. Assume W0
ǫ(µFωR) ≤ π and ∆ωRNR ·W
0
ǫ(µFωR) ≤
π
2. Then Wǫ0(µFωR) is
bounded as
cos ∆ωRNR·W
0 ǫ(µFωR)
≤pǫ(1−ǫ) +√ǫ.
Proof. Forǫ satisfying pǫ(1−ǫ) +√ǫ≥1 the claim is trivial and therefore we assume otherwise. In the following W0
ǫ(µFωR) is denoted by W
0
ǫ for notational simplicity. As stated above, the definition of the overall width around 0 implies ωR(F(I(0, W0
ǫ))) = tr[ρRF(I(0, W0
ǫ))] = 1 − ǫ, where ρR is a density operator representing ωR, i.e.,
ωR(A) = tr[ρRA] for allA∈ L(HR). Now we define shifted states (as density operators) of ρR by ρR,θ := e−iNRθρReiNRθ for −π ≤ θ ≤ π. Let us consider the fidelity [25, 26] between ρR and ρR,−W0
ǫ.
The fidelity between two states ρ0 and ρ1 is defined by F(ρ0, ρ1) := trh ρ1/20 ρ1ρ1/20 1/2i
, which takes values in [0,1]. It quantifies the closeness of two states such that F(ρ0, ρ1) = 1 holds if and only if ρ0 = ρ1. We note that the fidelity has another representation [27],
F(ρ0, ρ1) = inf
E:pom
X
x
tr[ρ0E({x})]1/2tr[ρ1E({x})]1/2,
where inf is taken over all discrete poms.
With I(0, W0
ǫ)c := [−π, π)\ I(0, Wǫ0), we have tr
ρRF(I(0, W0 ǫ)c)
= ǫ. Further, noting that the “shifted” (modulo 2π) set I(0, W0
ǫ)c −Wǫ0 = I(−Wǫ0, Wǫ0)c contains I(0, W0
ǫ) \ {−Wǫ0/2} (since Wǫ0 ≤ π), we also obtain tr
ρR,−W0
ǫF(I(0, W
0 ǫ)c)
= tr
ρRF(I(−W0
ǫ, Wǫ0)c)
≥tr
ρRF(I(0, W0 ǫ))
= 1−ǫ, and thus tr[ρR,−W0
ǫF(I(0, W
0 ǫ))]≤ ǫ. Therefore we have
F(ρR, ρR,−W0
ǫ)≤tr[ρRF(I(0, W
0
ǫ))]1/2tr[ρR,−W0
ǫF(I(0, W
0 ǫ))]1/2 + tr[ρRF(I(0, W0
ǫ)c)]1/2tr[ρR,−W0
ǫF(I(0, W
0 ǫ)c)]1/2 ≤pǫ(1−ǫ) +√ǫ.
The Mandelstam-Tamm uncertainty relation [28, 29] claims that for an arbitrary normalized vector |φi ∈ HR and θ with ∆φNR · |θ| ≤ π/2, |
φ|e−iNRθ
|φ
| ≥ cos (∆φNR·θ), where ∆φNR = hφ|N2
R|φi − hφ|NR|φi2
1/2
between general states ρ0 and ρ1 is also given as F(ρ0, ρ1) = sup|hφ0|φ1i| where sup is taken over all the possible purifications of ρ0 and ρ1, the Mandelstam-Tamm inequality is applied to purified states to give [30],
F(ρR, ρR,−θ)≥cos (∆ωRNR·θ).
Thus we obtain
cos ∆ωRNR·W
0 ǫ
≤pǫ(1−ǫ) +√ǫ.
Theorem 1. Let A be an effect defined by A = 12(|0ih0|+|1ih1|+|0ih1|+|1ih0|). For
ωR satisfying ∆ωRNR <
1 6,
D(A,ΓωR(U(A))) > 1 32.
For ωR satisfying∆ωRNR ≥
1
6, it holds D(A,ΓωR(U(A))) ≥
1 32
1−cos
π 12∆ωRNR
.
Proof. We put ǫ = 1
16. There are three possibilities: (i) W 0
1 16
(µF
ωR) ≤ π and ∆ωRNR · W
0
1 16
(µF
ωR) >
π
2, (ii) W 0
1 16
(µF
ωR) ≤ π and ∆ωRNR · W
0
1 16
(µF
ωR) ≤
π 2, (iii) W0
1 16
(µF
ωR)> π. (i) implies ∆ωRNR >
1
2. In this case, Proposition 2 gives
D(A,ΓωR(U(A))) ≥ 1 32
1−cos
π 4∆ωRNR
. (3)
Under condition (ii) lemma 2 applies. Forǫ= 161, pǫ(1−ǫ) +√ǫ≤2√ǫ= 12 holds and thus
∆ωRNR·W
0
1 16(µ
F
ωR)≥
π
6 (4)
follows. It implies ∆ωRNR≥
1 6 and
D(A,ΓωR(U(A))) ≥ 1 32
1−cos
π 12∆ωRNR
.
Combining (i) and (ii), we observe that ∆ωRNR <
1
6 implies W 0
1 16
(µF
ωR)> π, thus (iii). In this case, Proposition 2 is applied to show that
D(A,ΓωR(U(A))> 1
32. (5)
This completes the proof.
There are various reasons for going beyond these results. U relies on covariant
phases, and does not yield all invariant observables in H, immediately pointing to the need to understand the general case beyond the particular construction. We now present a model-independent trade-off between the operational measure of distance D(Γ(E), A) of an arbitrary effectA from the restriction of an invariant effectE, and the size of the reference system.
5. General argument
In this section we provide a general operational trade-off relation. We consider an arbitrary effect A ∈ E(HS) and a relative (invariant) effect E ∈ E(HS ⊗ HR) = E(H) and try to bound D(A,ΓωR(E)). Once more there is a symmetry (Lie) groupG acting (via projective unitary representations) both in L(HS) and L(HR), and we recall that for each element n in the Lie algebra g of G, there exist corresponding self-adjoint operators NS and NR such that for sufficiently small |s|, αS
ens(A) = e
iNSsAe−iNSs and
αR
ens(A) =e
iNRsAe−iNRs.
For notational simplicity we write Γ : L(H) → L(HS) for a channel of the form Γ = ΓωR for some ωR. The following is our main result.
Theorem 2. Recall that V(A) = kA −A2k, D(A, B) = kA−Bk, N
S and NR are
number operators on HS and HR respectively, and ωR ∈ S(HR). Then the following inequality holds:
[A, NS]
≤2D Γ(E), A
kNSk+ 2 ωR(NR2)−ωR(NR)21/2
2D(Γ(E), A) +V(A)1/2
.
This inequality shows that good proximity between a symmetric, relative observable and a non-symmetric absolute quantity requires a large spread in the reference system’s number operator.
Before we prove the theorem, we recall some relevant results concerning channels Λ : L(H) → L(K). In the application of the proof, we will specify Λ = Γ : L(HS ⊗ HR)→ L(HS).
Define a sesquilinear mapping h·,·i:L(H)× L(H)→ L(K) by
hA, Bi:= Λ(A∗B)−Λ(A∗)Λ(B).
This mapping satisfies hA, Ai ≥0 andhA, Bi∗ =hB, Ai for all A, B ∈ L(H). Thus this can be regarded as an “operator-valued inner product”. It satisfies a Cauchy-Schwarz inequality proven by Janssens [33], which we recapitulate here as we expect it to be useful for future steps in this line of investigation.
Lemma 3. Consider the map h·,·i defined above. For any A, B ∈ L(H) , it satisfies
Proof. We consider a (not necessarily minimal) Stinespring representation (H′, V) of the channel Λ. That is, H′ is a Hilbert space and V :K → H ⊗ H′ is an isometry such that Λ(A) =V∗(A⊗
1
H′)V holds. By introducing the operatorξ :=√1−V V∗, we can
represent the map as
hA, Bi=V∗(A∗ ⊗1
H′)ξ∗ξ(B⊗1
H′)V.
We then have
hA, BihB, Ai=V∗(A∗ ⊗1
H′)ξ∗ξ(B⊗1
H′)V V∗(B∗⊗1
H′)ξ∗ξ(A⊗1
H′)V.
As ξ(B ⊗1
H′)V V∗(B∗⊗1
H′)ξ∗ ≥0, it follows that
V∗(A∗⊗1H′)ξ
∗ξ(B⊗
1H′)V V
∗(B∗⊗
1H′)ξ
∗ξ(A⊗
1H′)V
≤ kξ(B⊗1
H′)V V∗(B∗⊗1
H′)ξ∗kV∗(A∗⊗1
H′)ξ∗ξ(A⊗1
H′)V.
Using the C∗ property of the norm, we obtain
kξ(B⊗1
H′)V V∗(B∗⊗1
H′)ξ∗k=kV∗(B∗⊗1
H′)ξ∗ξ(B⊗1
H′)Vk=khB, Bik,
which proves the lemma.
The following lemma is an immediate consequence of the previous result.
Lemma 4. For any A, B ∈ L(H),
khA, Bik2 ≤ khB, BikkhA, Aik.
We now have the following result.
Lemma 5. If A, B ∈ L(H) satisfy [A, B] = 0, then
k[Λ(A),Λ(B)]k ≤ kΛ(A∗A)−Λ(A)∗Λ(A)k1/2kΛ(BB∗)−Λ(B)Λ(B)∗k1/2
+kΛ(AA∗)−Λ(A)Λ(A)∗k1/2kΛ(B∗B)−Λ(B)∗Λ(B)k1/2. (6)
Proof. Appealing again to the sesquilinear mapping hA, Bi, we write
hA∗, Bi= Λ(AB)−Λ(A)Λ(B), hB∗, Ai= Λ(BA)−Λ(B)Λ(A).
Thus for A, B ∈ L(H) satisfying [A, B] = 0, it holds that
[Λ(A),Λ(B)] =hB∗, Ai − hA∗, Bi.
By Lemma 4, the result is proved.
Proof. Define ǫ := Γ(E)−A; we wish to estimate kǫk = D(Γ(E), A). Now we have [A, NS] = [Γ(E), NS] + [NS, ǫ] and thus
k[A, NS]k ≤ k[Γ(E), NS]k+k[NS, ǫ]k. (7) We provide a bound for each term. The second term of the right-hand side is easily bounded as k[NS, ǫ]k ≤2kNSkkǫk.
By assumption we have [E, N] = 0, and therefore we may apply Lemma 5 in order to bound the first term on the right hand side of (7). Note that Γ(N) =NS+ωR(NR)1.
We thus obtain
[Γ(E), NS]
=
[Γ(E),Γ(N)]
≤2
Γ(E2)−Γ(E)2
1/2
Γ(N2)−Γ(N)2
1/2
= 2Γ(E2)−Γ(E)2
1/2
ωR(NR2)−ω(NR)21/2
. (8)
We now boundkΓ(E2)−Γ(E)2k. Using Γ(E2)−Γ(E)2 ≥0 (two-positivity) andE2 ≤E, we obtain
Γ(E2)−Γ(E)2 ≤Γ(E)−Γ(E)2 = (ǫ+A)−(ǫ+A)2
=ǫ−ǫ2 −ǫA−Aǫ+A−A2.
Thus it holds that
kΓ(E2)−Γ(E)2k ≤ kǫ−ǫ2−ǫA−Aǫk+V(A) =
[A, ǫ] + (1−2A−ǫ)ǫ
+V(A)
≤[A, ǫ]
+
1−(A+ Γ(E))
ǫ+V(A).
0 ≤ A ≤ 1 gives
[A, ǫ]
≤ kǫk. In fact, [A, ǫ]
= supkφk=1| hφ|i[A, ǫ]|φi |
can be bounded by Robertson’s uncertainty relation as | hφ|i[A, ǫ]|φi | ≤
2
q
hφ|A2|φi − hφ|A|φi2kǫkwithqhφ|A2|φi − hφ|A|φi2 ≤ 1
2, and 0≤A+ Γ(E)≤ 21 gives
1−(A+ Γ(E))
≤1. Thus we obtain
Γ(E2)−Γ(E)2
≤2kǫk+V(A). (9)
Putting (9) in (8), we obtain
k[Γ(E), NS]k ≤2(2kǫk+V(A))1/2(ωR(NR2)−ω(NR)2)1/2. (10)
Thus, (7) can be bounded as
[A, NS]
≤2kNSkkǫk+ 2 2kǫk+V(A) 1/2
ωR(NR2)−ω(NR)21/2
.
This completes the proof.
Corollary 1. It holds that
[A, NS]
≤2D(Γ(E), A)kNSk+ 2kNRk 2D(Γ(E), A) +V(A) 1/2
.
Corollary 2. For a sharp effect (projection) A it holds that
[A, NS]
≤2D(Γ(E), A)kNSk+ 2
√
2 ωR(NR2)−ωR(NR)21/2
D(Γ(E), A)1/2,
Thus to attain a good measurement a large reference frame is required, since it is only then that ωR(N2
R)−ωR(NR)2 can be large.
Example 1. (G=U(1)). In section 4 we studied the particular mapU for G=U(1)
and showed that a large reference frame is needed for U(A) to well approximate the
effect A = 1
2(|0ih0|+|1ih1|+|0ih1|+|1ih0|). Below we estimate the lower bound of the distance D(Γ(E), A) for an arbitrary invariantE. Since A acts only on a subspace spanned by {|0i,|1i}, we introduce a projection operator P = |0ih0|+|1ih1|. We first note that for an arbitrary invariantE, we have
D Γ((P ⊗1)E(P ⊗1)), A
≤D Γ(E), A
.
In fact, since Γ((P ⊗1)E(P ⊗1)) = PΓ(E)P holds,§ D Γ((P ⊗1)E(P ⊗1)), A
=
P(Γ(E)−A)P
≤ D(Γ(E), A) follows. Thus it suffices to consider only E satisfying
(P⊗1)E(P⊗1) =E. For such an E,ǫ:= Γ(E)−A satisfies [N
S, ǫ] = [P NSP, ǫ]. Thus
[NS, ǫ]
in (7) can be bounded as [NS, ǫ]
≤ kǫk (where positivity of NS is used).
Compared to the direct application of Corollary 2, this gives the tighter bound 1
2 ≤D(ΓωR(E), A) + 2
√
2(ωR(NR2)−ωR(NR)2)1/2D(ΓωR(E), A)
1/2,
which shows a trade-off relation between the accuracy and the size of the reference frame.
Discussion. The necessity of the large reference frame, in the sense of large fluctuation
ωR(N2
R)−ωR(NR)2, can be interpreted in terms of the uncertainty relation for joint measurability [20]. E commutes with N := NS +NR, and one may consider joint measurements of E and N where these observables may be regarded as approximators to A and NS, respectively. The aim is to obtain outcome distributions of E that are close to those of A. Since A and NS do not commute, this closeness entails, due to the joint measurement uncertainty relation, that the outcome distribution of N should contain less “information” on NS. In other words, the (hypothetical) measurement of N must be a highly inaccurate measurement of NS. This large approximation error can be achieved only by the initial uncertainty of NR. Put another way, for A to be a good “absolute” representative of E in the context of a joint measurement ofE and N, by the uncertainty relation for A and NS, NS cannot well approximate N, and this discrepancy between NS and N is afforded by a large spread in NR.
We also note that symmetry constraints of the form we have considered are relevant in the framework of resource theories (e.g., [16], [34]), wherein a reference state ωR
is viewed as a resource if it is not invariant under U(1) symmetry transformations. The asymmetric state is seen in [16] as allowing for the simulation of non-invariant statistics ofS under the constraint of symmetric operations ofS+R, and the degree of asymmetry may be quantified and exploited in the Wigner-Araki-Yanase theorem [34]. The connection between asymmetry and localisation, as we have it, is clear: highly localised reference states, with respect to a phase conjugate to number, are highly asymmetric under phase shifts. The exact equivalence, should there be one, between localisation as allowing non-invariant quantities to represent relative ones on the one hand, and asymmetry as a resource for good measurements in Wigner-Araki-Yanase-type scenarios on the other, remains a task to be investigated.
6. Conclusion
Through model considerations and generic trade-off relations we have shown that the possibility of the traditional tacit reduction of relative quantities to “absolute” quantities is contingent upon the size of the reference system and a judicious choice of reference (system) state. An obstruction to complete specification of subsystem statistics, or rather to subsystem quantities being used to approximate the corresponding invariant quantities, arises due to the incompatibility of subsystem quantities.
The necessity of a large uncertainty in the symmetry generator in a given reference state for good approximation is an essentially quantum feature. In classical mechanics, all observables commute and all (pure) states are well localised with respect to all classical quantities. Provided that the classical reference system admits a faithful action of the symmetry group, this reference system is sufficient for the “absolute” quantities to perfectly represent the relative ones, with no constraint on the values of other quantities at all.
The analysis presented here constitutes a first step towards a comprehensive, operational understanding of the role of (quantum) reference systems in the description of quantum experiments, and suggests a number of further avenues of exploration, for example the physically relevant case of non-Abelian symmetries, approximation of invariant observables by “absolute” phase-space quantities, strength of Bell inequality violation for finite-size reference systems, and many more.
Acknowledgements Many thanks to Tom Bullock for a careful reading of an earlier version of this manuscript. TM acknowledges JSPS KAKENHI (grant no. 15K04998). LL acknowledges support under the grantQuantum Mathematics and Computation (no. EP/K015478/1).
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